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Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture

Yuhao Cheng

Abstract

We continue generalizing Altuğ's work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the unramified setting for \emph{Beyond Endoscopy} to the ramified case where ramification occurs at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$, after generalizing the first step. We establish a new proof of the $1/4$ bound towards the Ramanujan conjecture for the trace of the cuspidal part in the ramified case, which is also provided by adapting Altuğ's original approach. The proof proceeds in three stages: First, we estimate the contributions from the non-elliptic parts of the trace formula. Then, we apply the main result from our the previous work to isolate the $1$-dimensional representations within the elliptic part. Finally, we employ technical analytic estimates to bound the remainder terms in the elliptic part.

Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture

Abstract

We continue generalizing Altuğ's work on over in the unramified setting for \emph{Beyond Endoscopy} to the ramified case where ramification occurs at with , after generalizing the first step. We establish a new proof of the bound towards the Ramanujan conjecture for the trace of the cuspidal part in the ramified case, which is also provided by adapting Altuğ's original approach. The proof proceeds in three stages: First, we estimate the contributions from the non-elliptic parts of the trace formula. Then, we apply the main result from our the previous work to isolate the -dimensional representations within the elliptic part. Finally, we employ technical analytic estimates to bound the remainder terms in the elliptic part.

Paper Structure

This paper contains 27 sections, 59 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The main result in this paper and proof strategy
  4. Notations
  5. Preliminaries
  6. The modified $p$-adic norm and properties
  7. The elliptic part of the trace formula and measure normalizations
  8. Singularities of the orbital integrals
  9. Nonarchimedean case
  10. Archimedean case
  11. Global notations
  12. The partial Zagier $L$-function and the approximate functional equation
  13. The generalized Kloosterman sum
  14. Contribution of non-elliptic parts
  15. Estimates of the geometric side
  16. ...and 12 more sections

Key Result

Theorem 1.1

For any $\varepsilon>0$ and $f^n$ as above, we have where the implied constant only depends on $f_\infty$, $f_{q_i}$ and $\varepsilon$.

Theorems & Definitions (66)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6: Functional equation of $L^{S}(s,\delta)$, cheng2025 Theorem 3.3
  • Remark 2.7
  • Theorem 2.8: Approximate functional equation for $L^{S}(s,\delta)$, cheng2025 Theorem 3.8
  • ...and 56 more